vi PREFACE
Chapter
3 deals
with
the
two basic
number
systems,
the
ordinal num-
bers,
and
the
cardinal numbers.
The
arithmetics of
both
systems are de-
veloped sufficiently
to
allow for most applications outside set theory.
In
Chapter
4,
I delve into
the
subject
set
theory
itself. Since contem-
porary
set
theory
is
a very large subject, this foray
is
of necessity very
restricted. I have two aims in including it. First,
it
provides good examples
of
the
previous theory.
And
second,
it
gives
the
reader some idea of
the
flavor of
at
least some
parts
of
pure
set theory.
Chapter
5 presents a modification of Zermelo-Fraenkel set theory.
The
Zermelo-Fraenkel system has a
major
defect as a foundational subject.
Many easily formulated problems
cannot
be solved in
the
system.
The
Axiom of Constructibility is
an
axiom
that,
when added
to
the
Zermelo-
Fraenkel system, eliminates most, if
not
all, of these undecidable problems.
In
Chapter
6,
I give
an
account of
the
method
by which one
can
prove
within
the
Zermelo-Fraenkel system
that
various
statements
are themselves
not
provable in
that
system.
Chapters
5
and
6 are nonrigorous. My aim is
to
explain
rather
than
develop.
They
are included because of
their
relevance
to
other
areas of
mathematics. A detailed investigation of these topics would double
the
length of
this
book
at
the
very least
and
as such is
the
realm of
the
set-
theorist,
though
I would,
of
course,
be
delighted
to
think
that
any of my
readers would
be
encouraged
to
go further into these
matters.
Finally,
in
Chapter
7,
I present
an
introductory
account of
an
alternative
conception of set
theory
that
has proved useful in computer science (and
elsewhere),
the
non-well-founded set
theory
of
Peter
Aczel.
Chapters
1
through
3 contain numerous easy exercises.
In
Chapters
1
and
2,
they
are formally designated as 'Exercises'
and
are intended for
solution as
the
reader proceeds.
The
aim is
to
provide enough material
to
help
the
student
understand
fully
the
concepts
that
are introduced.
In
Chapter
3,
the
exercises
take
the
form of simple proofs of basic lemmas,
which are left
to
the
reader
to
provide. Again,
the
aim is
to
assist
the
reader's comprehension.
At
the
end
ofeach of
Chapters
1
through
3, there is also a small selection
of problems. These are more challenging
than
the
exercises
and
constitute
digressions from,
or
extensions of,
the
main
development.
In
some instances
the
reader
may
need
to
seek assistance in order
to
do these problems.
This
book is a
greatly
expanded second edition of my earlier Fundamen-
tals
of
Contemporary
Set
Theory, published by Springer-Verlag in 1979.
In
addition
to
the
various changes I have
made
to
my original account, I could
not
resist a change in title, relegating
the
title
of
the
first edition
to
a sub-
title
for
the
second,
thereby
enabling me
to
join
the
growing ranks of Joy
